Chen, Ming-chieh

Shieh, Yeung-Nan

English

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San Jose State UniversitySJSU ScholarWorksFaculty Publications Economics1-1-2009The Effects of Specific Commodity Taxes onOutput and Location of Free Entry OligopolyYeung-Nan ShiehSan Jose State University, yeung-nan.shieh@sjsu.eduMing-chieh ChenFollow this and additional works at: https://scholarworks.sjsu.edu/econ_pubPart of the Economics CommonsThis Article is brought to you for free and open access by the Economics at SJSU ScholarWorks. It has been accepted for inclusion in FacultyPublications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.Recommended CitationYeung-Nan Shieh and Ming-chieh Chen. "The Effects of Specific Commodity Taxes on Output and Location of Free Entry Oligopoly"Economics Bulletin (2009): 3104-3111.           Volume 29, Issue 4   The effects of specific commodity taxes on output and location of free entry oligopoly     Ming-chieh Chen  Jinwen University of Sience & Technology Yeung-nan Shieh  San Jose State UniversityAbstract This paper examines the impact of a specific commodity tax on output and the location decision of undifferentiated oligopolistic firms with free entry. It shows that (1) the optimum output and location of the oligopolistic firm is independent of the specific commodity tax if the demand function is linear (2) an increase in the specific commodity tax will increase (decrease) output per firm and move the plant location toward (away from) the output market if the demand function is concave (convex). These results are consistent with the conventional results based on the non-spatial setting. In the case in which the demand function is linear or concave, it shows that the number of firms and total output of oligopoly may increase. These results are significantly different from the conventional results based on non-spatial setting. It indicates that the effects of the specific tax on total output and the number of firms crucially depend upon transport costs and the location decisions of oligopolistic firms.We are grateful to Professor Xinghe Wang for valuable comments. All remaining errors are ours. Citation: Ming-chieh Chen and Yeung-nan Shieh, (2009) ''The effects of specific commodity taxes on output and location of free entry oligopoly'', Economics Bulletin, Vol. 29 no.4 pp. 3104-3111. Submitted: Feb 26 2009.   Published: December 12, 2009.                            1. Introduction In his famous paper, Commodity Taxation and Imperfect Competition (1989), Besley investigated the effects of a commodity tax on output per firm, the number of firms and total output of undifferentiated oligopolistic firms with free entry.  Under the assumptions that (1) firms are symmetric and identical; (2) firms produce a homogenous good and make Cournot-Nash conjectures about their rivals’ production decisions; (3)firms are free to enter and leave the industry; (4) the sufficient second order conditionsand the stability conditions are satisfied, Besley obtained the following propositions. B1. An increase in the specific commodity tax will increase (decrease) output per firm if the inverse demand function is concave (convex). Besley (1989, p. 363). B2. An increase in the specific commodity tax will decrease the number of firms if the demand function is linear or concave. Besley (1989, p. 363). B3. An increase in the specific commodity tax will decrease total output. Besley (1989, p. 363). These results are based on the non-spatial setting in which location and transport costs are negligible.  However, the real economy is characterized by dispersion of consumers and producers over geographic space with trade between them always incurring transport costs. It would be interesting and important to investigate the effects of the specificcommodity tax on output per firm, the number of firms and total output ofundifferentiated oligopolistic firms in a spatial setting. The purpose of this paper is to fill this gap.  It explicitly incorporates oligopolisticmarket structure into the Weber triangle and examines the impact of a change in the specific commodity tax on output and the plant location of undifferentiated oligopolistic firms.  It will be shown that the well-known Besley’s B1 holds but B2 and B3 may not hold in the oligopolistic location model. 2. An Oligopolistic Location Model Our analysis is based on the well-known Weber triangular model with the following assumptions. (a) N firms employ two transportable inputs (m1 and m2) located at A and B to produce a homogenous product (Q) which is sold at the output market locating at C.  The location triangle in Figure 1 illustrates the location problem of oligopolistic firms.  In figure 1, the distance a and b and the angle γ are known; h is the distance between the plant location (E) and the output market (C); z1 and z2 are the distances of plant location (E) from A and B, respectively; θ is the angle between CA and CE. (b) Firms make Cournot-Nash conjectures about their rivals’ production and location decisions and enter the industry without any restrictions until there is no economicprofit. Assume also that equilibra are symmetric.  Thus, we can neglect the location dispersion of firms and focus on the impact of the specific commodity tax on the production and location decisions of a representative firm. 1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                C γ I θ  J h a E b z1  z2 A B Figure 1. The Weber Triangle (c) The production function is homothetic and can be specified as:       q = f(m1, m2) 	 (1) with fm1 ≡ ∂q/∂m1 > 0, fm2 ≡ ∂q/∂m2 > 0, fm1m1 ≡ ∂2q/∂m1 2 < 0, and fm2m2 ≡ ∂2q/∂m2 2 < 0. (d) The industry inverse demand function for output is given by P = P(Q) 	 (2) where Q = ∑qi is the market quantity demanded,  PQ ≡ ∂P/∂Q < 0, PQ + qPQQ < 0, N    cf. HM (1992, p. 256).  It should be noted that ∑ denotes ∑  . i=1 (e) The prices of inputs and output are evaluated at the plant location (E).  	The cost of purchasing inputs is the price of input at the source plus the freight cost, and the price of output is the market price minus the freight cost. (f) Transportation rates are constant. (g) The government imposes a specific tax which can be specified as: T = tq 	 (3) where t = the specific tax rate, 1 > t > 0. (h) The objective of each firm is to find the optimum location and production within the Weber triangle which maximizes the profit. With these assumptions, the profit maximizing location problem of the representative firm is given by     max Π = [P(Q)-rh]f(m1, m2) – (w1+r1z1)m1 – (w2+r2z2)m2 – tf(m1, m2) (4) 2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             where z1 = (a2 + h2 – 2ahcosθ)1/2, z2 = [b2 + h2 – 2bhcos(-θ)]1/2; w1 and w2 are the base prices of m1 and m2 at their sources A and B; r, r1 and r2 are constant transportation ratesof q, m1, m2; z1, z2, and h are the distances from the plant location to the source location A, B and the market location C.  It is worth mentioning that q, m1, m2, h and θ are choice variables and a, b, t, , w1, w2, r, r1, r2 are positive parameters.      Assuming that the oligopolistic firm treats q instead of m1 and m2 as a decision variable, we first derive the cost function by minimizing total cost subject to a given output at a given location,     min L = (w1+r1z1)m1 – (w2+r2z2)m2  + [q – f(m1, m2)] (5) where   is the Lagrange multiplier; q, h and θ are parameters.  Using the standard comparative static analysis and the envelope theorem, we can show that the production function is homothetic if and only if the production cost function is separable in the sensethat C(q; h, θ) = c(w1+r1z1, w2+r2z2)H(q) (6) where c is a function of the delivered prices of m1 and m2, e.g. Takayama (1993, Proposition 3.5., pp. 147-148). Hence, the average cost and marginal cost can be written as: AC = C(q; h, )/q = c(.)H(q)/q (7) MC = Cq = c(.)Hq  (8) where Cq ≡ ∂(q; h, )/∂q and Hq ≡  dH(q)/dq.      Following Hanoch (1975), from (7) and (8), we obtain the following relation: H(q)/q > (=) < Hq  (9) if the production function exhibits increasing (constant) or decreasing returns to scale. Substituting the production cost function C = C(q; h, ) into (4), we obtain the profit as a function of q, θ and h. The first-order condition for a maximum would be ∂Π/∂q = [(P + PQq) – rh] – c(.)Hq – t = 0 (10) ∂Π/∂θ = - cθH(q) = 0 (11) ∂Π/∂h = - rq – chH(q) = 0 (12) where cθ ≡ ∂c(.)/∂θ, ch ≡ ∂c(.)/∂h. Assume that the second-order conditions are satisfied and the possibility of the corner solution is excluded; cf. Kusumoto (1986) and Mai and Hwang (1992). We can solve (10)-(12) for q, θ and h when free entry is prohibited. If free entry is allowed, each firm in the industry earns normal profit only.  The following condition must be satisfied. Π = [P(Nq) – rh]q – c(.)H(q) – tq = 0 (13) 3                                                                                                                                                                                                                                                                                                                                                                                                    If there is an interior solution, we can solve equations (10) – (13) for q, θ, h and N in terms of t and v = (a, b, γ, w1, w2, r1, r2, r), where v is a vector of remaining parameters. q = q(t, v), θ = θ(t, v), h = h(t, v), N = N(t, v) (14) The expressions for the partial derivatives such as ∂q/∂t, ∂θ/∂t, ∂h/∂t and ∂N/∂t can beobtained by applying the standard comparative static analysis.  It is of interest to note that the production function must exhibit increasing returns to scale for (10) – (13) to have a solution as in (14). To see this, we divide both sides of equation (13) by q and obtain [P(Nq) – rh] = [c(.)H(q)/q] – t (15) Substituting (15) into (10), we obtain PQq = c(.)[Hq(q) - H(q)/q] (16) Since the left-hand side of (16) is negative, for the right-hand side of (16) to be negative, the production function must exhibit increasing returns to scale, i.e., H(q)/q > Hq(q) (seealso Hwang, Mai and Shieh, 2007) . It simply implies that in equilibrium all firms produce on the downward sloping part of the average cost curve under Cournot-Nash competition with free entry.         This completes our modeling of the basic framework for studying the effects of aspecific tax on the oligopolistic firm’s production and location decisions. 3. Effects of Specific Taxes on Production and Location Decisions We are now in a position to examine the effects of a change in the specific tax rate on theoptimum output and location.  Totally differentiating equations (10)-(13) and applying Cramer’s rule, we obtain the following results. (∂θ/∂t) = (-1/D4)PQQq3Πθhch{[H(q)/q] – Hq} (17) (∂h/∂t) = (1/D4)PQQq3Πθθch{[H(q)/q] – Hq} (18) (∂q/∂t) = (-D2/D4)PQQq3  (19) (∂N/∂t) = (q/D4)(D2{[2PQ + PQQq – c(.)Hqq] + (N - 1)PQQq} – ΠθθΠqh 2) (20) (∂Q/∂t) = N(∂q/∂t) + q(∂N/∂t) = (q/D4){D2q[2PQ – c(.)Hqq] – ΠθθΠqh 2} (21) where Πθh = - chθH(q), Πθθ = - cθθH(q), Πqq = (N + 1)PQ + NPQQq – cHqq, Πq = PQq(N – 1), Πqh = ch{[H(q)/q] – Hqq}, D2 = ΠθθΠhh - Πθh 2 and D4 is the relevant Hessiandeterminant.  It should be noted that Πθθ < 0, D2 > 0 and D4 > 0 by the stabilityconditions, ch < 0 can be seen from equation (12) and [H(q)/q] – Hq > 0 is due to increasing returns to scale. It is clear that the signs of (∂θ/∂t), (∂h/∂t) and (∂q/∂e) crucially depend upon the shape of market demand function.  In the case where the market demand function is linear, i.e., 4                                                                                                                                                                                                PQQ = 0. From (17) – (19), we obtain (∂q/∂t) = 0, (∂θ/∂t) = 0 and (∂h/∂t) = 0. Thus, wecan conclude that Proposition 1. The optimum output and location of an oligopolistic firm is independent of a change in the specific tax if the demand function is linear.The economic interpretation behind Proposition 1 is given as follows.  A change in the specific tax does not change the slope of the demand curve at any output level but will increase the output price in equilibrium for the oligopolistic firms to break even.  In the case where the demand function is linear, i.e., PQQ = 0, a higher output price will not alter the slope of demand curve and so the required tangency between demand curve and average cost curve occurs at the same output level for each firm, i.e., (∂q/∂t) = 0. Since the output per firm remains unchanged, the optimum location will remain the same.      Next, we consider the case where the demand function is not linear, i.e., PQQ ≠ 0. Since the signs of PQQ and Πθh can not a priori be determined, the signs of (∂q/∂t), (∂θ/∂t) and (∂h/∂t) in (17) - (19) are ambiguous.  However, from (18) and (19), we can obtain  (∂q/∂t) < ( >) 0, as PQQ > (<) 0 (22) (∂h/∂t) > (<) 0, as PQQ > (<) 0 (23) Thus, we can conclude that Proposition 2.  An increase in the specific tax will increase (decrease) the output of an oligopolistic firm and will move its plant location closer to (farther away from) the CBD if the demand function is concave (convex).The impact of the specific commodity tax on the output per firm is consistent with B1. The economic intuition underlying Proposition 2 is given as follow.  An increase in the specific tax rate does not change the slope of the demand curve at any output level but will increase the output price in equilibrium for firms to break even.  In the case where the demand function is concave (i.e., PQQ < 0), a higher output price decreases the absolute value of the slope of the demand curve and so the point of tangency between demand curve and average curve occurs at a larger output level for each firm.  Since the production function exhibits increasing returns to scale, the quantity of inputs, m1 and m2, per unit of output declines, then the resources pull decreases while the market pull increases. As a result, the optimum location moves towards the CBD.  In the case where the demand function is convex (i.e., PQQ > 0), the opposite applies. Next, we turn to the effect of a change in the specific tax on equilibrium number of undifferentiated oligopolistic firms.  From equation (20), we can see the sign of (∂N/∂t)can not be a priori determined because [2PQ + PQQq – c(.)Hqq]D2 < 0, - ΠθθΠqh 2 > 0 and the sign of (N-1)PQQq2 can be either positive or negative depending on the shape of demand function.  We can show (∂N/∂t) > (<) 0, as - D2{[2PQ + PQQq – c(.)Hqq] + (N - 1)PQQq} > (<) – ΠθθΠqh 2  (24) Thus, we can conclude that 5                                    Proposition 3. An increase in the specific commodity tax may increase the number of undifferentiated firms even if the demand function is linear or concave. This result is significantly different from that of Besley (1989, p. 363) in the non-spatial setting. The different results are due to the location effect, ΠθθΠqh 2 = - cθθH(q) {ch[H(q)/q]-Hq}2 . It shows that the firm will change its plant location after the change of the specific commodity tax.  If the location effect dominates the output effect, the number of firms may increase. Finally, we consider the effect of an increase in the commodity specific tax on the total output of oligopoly. From equation (21), we obtain (∂Q/∂t) < (>) 0, as - D2q[2PQ – c(.)Hqq] > (< ) – ΠθθΠqh 2  (25) Thus, we can conclude that Proposition 8. An increase in the specific tax may increase total output of undifferentiated oligopoly.This result is also significantly different from that of Besley (1989, p. 363).  Once again the different results are due to the location effect.   4. Concluding Remarks We examine the impact of a specific commodity tax on the production and plant location decisions of undifferentiated oligopolistic firms with free entry.  In the case where the demand function is linear, we show that an increase in the specific commodity tax does not change the location decision and output of an oligopolistic firm. In the case where the demand function is not linear, we show that an increase in the specific tax will cause each firm’s output to rise (fall) and move the plant location closer to (farther away from)the output market if the demand function is concave (convex).  These results indicate that B1 holds in the Weber triangular location model. In the case where the demand function is linear or concave, we show that an increasein the specific commodity tax may increase the number of firms and total output of oligopoly. This result is significantly different from B2 and B3.  It indicates that the location decision has very important influence on the impact of a change in the specific commodity tax on the number of firms and total output of oligopoly with free entry. ReferencesBesley, T. (1989) “Commodity taxation and imperfect competition: a note on the effects of entry” Journal of Public Economics 40, 359-67. Hanoch, G. (1975) “The elasticity of scale and the shape of average costs” American Economic Review 65, 492-497. 6                    Hwang, H., Mai, C. C. and Shieh, Y. N. (2007) “Production-location decision and freeentry oligopoly: a correction” Journal of Urban Economics 68, 576-79. Kusumoto, S. I. (1984) “On a foundation of the economic theory of location - transport distance vs. technological substitution” Journal of Regional Science 24, 249-270. Mai, C. C. and Hwang, H. (1992) “Production-location decision and free entry oligopoly” Journal of Urban Economics 31, 252-71. Seade, J. (1980) “On the effects of entry” Econometrica 48, 479-89. Takayama, A. (1993) Analytical Methods in Economics, University of Michigan Press: Ann Arbor. Weber, A. (1929) Theory of Location of Industry, translated by C.J. Friedrich, Universityof Chicago Press: Chicago. 7 

The Effects of Specific Commodity Taxes on Output and Location of Free Entry Oligopoly

San Jose State University

San Jose State University
SJSU ScholarWorks
Faculty Publications Economics
1-1-2009
The Effects of Specific Commodity Taxes on
Output and Location of Free Entry Oligopoly
Yeung-Nan Shieh
San Jose State University, yeung-nan.shieh@sjsu.edu
Ming-chieh Chen
Follow this and additional works at: https://scholarworks.sjsu.edu/econ_pub
Part of the Economics Commons
This Article is brought to you for free and open access by the Economics at SJSU ScholarWorks. It has been accepted for inclusion in Faculty
Publications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.
Recommended Citation
Yeung-Nan Shieh and Ming-chieh Chen. "The Effects of Specific Commodity Taxes on Output and Location of Free Entry Oligopoly"
Economics Bulletin (2009): 3104-3111.
     
 
 
  
  
Volume 29, Issue 4 
  
The effects of specific commodity taxes on output and location of free entry 
oligopoly 
  
 
 
Ming-chieh Chen  
Jinwen University of Sience & Technology 
Yeung-nan Shieh  
San Jose State University
Abstract 
This paper examines the impact of a specific commodity tax on output and the location decision of undifferentiated 
oligopolistic firms with free entry. It shows that (1) the optimum output and location of the oligopolistic firm is 
independent of the specific commodity tax if the demand function is linear (2) an increase in the specific commodity 
tax will increase (decrease) output per firm and move the plant location toward (away from) the output market if the 
demand function is concave (convex). These results are consistent with the conventional results based on the non-
spatial setting. In the case in which the demand function is linear or concave, it shows that the number of firms and 
total output of oligopoly may increase. These results are significantly different from the conventional results based on 
non-spatial setting. It indicates that the effects of the specific tax on total output and the number of firms crucially 
depend upon transport costs and the location decisions of oligopolistic firms.
We are grateful to Professor Xinghe Wang for valuable comments. All remaining errors are ours. 
Citation: Ming-chieh Chen and Yeung-nan Shieh, (2009) ''The effects of specific commodity taxes on output and location of free entry 
oligopoly'', Economics Bulletin, Vol. 29 no.4 pp. 3104-3111. 
Submitted: Feb 26 2009.   Published: December 12, 2009. 
 
     
  
 
 
 
 
 
 
 
     
 
 
 
 
 
 
 
1. Introduction 
In his famous paper, Commodity Taxation and Imperfect Competition (1989), Besley 
investigated the effects of a commodity tax on output per firm, the number of firms and 
total output of undifferentiated oligopolistic firms with free entry.  Under the 
assumptions that (1) firms are symmetric and identical; (2) firms produce a homogenous 
good and make Cournot-Nash conjectures about their rivals’ production decisions; (3)
firms are free to enter and leave the industry; (4) the sufficient second order conditions
and the stability conditions are satisfied, Besley obtained the following propositions. 
B1. An increase in the specific commodity tax will increase (decrease) output per firm if 
the inverse demand function is concave (convex). Besley (1989, p. 363). 
B2. An increase in the specific commodity tax will decrease the number of firms if the 
demand function is linear or concave. Besley (1989, p. 363). 
B3. An increase in the specific commodity tax will decrease total output. Besley (1989, p. 
363). 
These results are based on the non-spatial setting in which location and transport costs are 
negligible.  However, the real economy is characterized by dispersion of consumers and 
producers over geographic space with trade between them always incurring transport 
costs. It would be interesting and important to investigate the effects of the specific
commodity tax on output per firm, the number of firms and total output of
undifferentiated oligopolistic firms in a spatial setting. 
The purpose of this paper is to fill this gap.  It explicitly incorporates oligopolistic
market structure into the Weber triangle and examines the impact of a change in the 
specific commodity tax on output and the plant location of undifferentiated oligopolistic 
firms.  It will be shown that the well-known Besley’s B1 holds but B2 and B3 may not 
hold in the oligopolistic location model. 
2. An Oligopolistic Location Model 
Our analysis is based on the well-known Weber triangular model with the following 
assumptions. 
(a) N firms employ two transportable inputs (m1 and m2) located at A and B to produce a 
homogenous product (Q) which is sold at the output market locating at C.  The 
location triangle in Figure 1 illustrates the location problem of oligopolistic firms.  In 
figure 1, the distance a and b and the angle γ are known; h is the distance between the 
plant location (E) and the output market (C); z1 and z2 are the distances of plant 
location (E) from A and B, respectively; θ is the angle between CA and CE. 
(b) Firms make Cournot-Nash conjectures about their rivals’ production and location 
decisions and enter the industry without any restrictions until there is no economic
profit. Assume also that equilibra are symmetric.  Thus, we can neglect the location 
dispersion of firms and focus on the impact of the specific commodity tax on the 
production and location decisions of a representative firm. 
1
 
  
     
                                                              
                                                              
                                                   
                                                  
   
                                                       
 
 
         
   
 
 
  
 
  
 
    
 
                                             
                                                                        
 
                                                                                                           
                
 C 

γ
 I θ  J 
h 
a E b 
z1  z2
 A B 
Figure 1. The Weber Triangle 
(c) The production function is homothetic and can be specified as: 
      q = f(m1, m2) 	 (1) 
with fm1 ≡ ∂q/∂m1 > 0, fm2 ≡ ∂q/∂m2 > 0, fm1m1 ≡ ∂2q/∂m1 2 < 0, and 
fm2m2 ≡ ∂2q/∂m2 2 < 0. 
(d) The industry inverse demand function for output is given by 
P = P(Q) 	 (2) 
where Q = ∑qi is the market quantity demanded,  PQ ≡ ∂P/∂Q < 0, PQ + qPQQ < 0, 

N 

   cf. HM (1992, p. 256).  It should be noted that ∑ denotes ∑  . 

i=1 

(e) The prices of inputs and output are evaluated at the plant location (E).  	The cost of 
purchasing inputs is the price of input at the source plus the freight cost, and the price 
of output is the market price minus the freight cost. 
(f) Transportation rates are constant. 
(g) The government imposes a specific tax which can be specified as: 
T = tq 	 (3) 
where t = the specific tax rate, 1 > t > 0. 
(h) The objective of each firm is to find the optimum location and production within the 
Weber triangle which maximizes the profit. 
With these assumptions, the profit maximizing location problem of the representative 
firm is given by 
    max Π = [P(Q)-rh]f(m1, m2) – (w1+r1z1)m1 – (w2+r2z2)m2 – tf(m1, m2) (4) 
2

  
 
 
 
 
 
 
 
 
                                                                                                                          
 
   
 
 
     
 
    
    
    
 
      
     
 
    
                                         
                                                                       
                                                                                
                                                              
                                                                                             
                                                                                     
where z1 = (a2 + h2 – 2ahcosθ)1/2, z2 = [b2 + h2 – 2bhcos(-θ)]1/2; w1 and w2 are the base 
prices of m1 and m2 at their sources A and B; r, r1 and r2 are constant transportation rates
of q, m1, m2; z1, z2, and h are the distances from the plant location to the source location 
A, B and the market location C.  It is worth mentioning that q, m1, m2, h and θ are choice 
variables and a, b, t, , w1, w2, r, r1, r2 are positive parameters. 
     Assuming that the oligopolistic firm treats q instead of m1 and m2 as a decision 
variable, we first derive the cost function by minimizing total cost subject to a given 
output at a given location, 
    min L = (w1+r1z1)m1 – (w2+r2z2)m2  + [q – f(m1, m2)] (5) 
where   is the Lagrange multiplier; q, h and θ are parameters.  Using the standard 
comparative static analysis and the envelope theorem, we can show that the production 
function is homothetic if and only if the production cost function is separable in the sense
that 
C(q; h, θ) = c(w1+r1z1, w2+r2z2)H(q) (6) 
where c is a function of the delivered prices of m1 and m2, e.g. Takayama (1993, 
Proposition 3.5., pp. 147-148). Hence, the average cost and marginal cost can be written 
as: 
AC = C(q; h, )/q = c(.)H(q)/q (7) 
MC = Cq = c(.)Hq  (8) 
where Cq ≡ ∂(q; h, )/∂q and Hq ≡  dH(q)/dq. 
     Following Hanoch (1975), from (7) and (8), we obtain the following relation: 
H(q)/q > (=) < Hq  (9) 
if the production function exhibits increasing (constant) or decreasing returns to scale. 
Substituting the production cost function C = C(q; h, ) into (4), we obtain the profit 
as a function of q, θ and h. The first-order condition for a maximum would be 
∂Π/∂q = [(P + PQq) – rh] – c(.)Hq – t = 0 (10) 
∂Π/∂θ = - cθH(q) = 0 (11) 
∂Π/∂h = - rq – chH(q) = 0 (12) 
where cθ ≡ ∂c(.)/∂θ, ch ≡ ∂c(.)/∂h. Assume that the second-order conditions are satisfied 
and the possibility of the corner solution is excluded; cf. Kusumoto (1986) and Mai and 
Hwang (1992). We can solve (10)-(12) for q, θ and h when free entry is prohibited. 
If free entry is allowed, each firm in the industry earns normal profit only.  The 
following condition must be satisfied. 
Π = [P(Nq) – rh]q – c(.)H(q) – tq = 0 (13) 
3
 
  
     
 
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
     
    
                                                                                 
                                                                                       
                                                          
                                                           
               
                                                   
 
If there is an interior solution, we can solve equations (10) – (13) for q, θ, h and N in 
terms of t and v = (a, b, γ, w1, w2, r1, r2, r), where v is a vector of remaining parameters. 
q = q(t, v), θ = θ(t, v), h = h(t, v), N = N(t, v) (14) 
The expressions for the partial derivatives such as ∂q/∂t, ∂θ/∂t, ∂h/∂t and ∂N/∂t can be
obtained by applying the standard comparative static analysis.  It is of interest to note 
that the production function must exhibit increasing returns to scale for (10) – (13) to 
have a solution as in (14). To see this, we divide both sides of equation (13) by q and 
obtain 
[P(Nq) – rh] = [c(.)H(q)/q] – t (15) 
Substituting (15) into (10), we obtain
 PQq = c(.)[Hq(q) - H(q)/q] (16) 
Since the left-hand side of (16) is negative, for the right-hand side of (16) to be negative, 
the production function must exhibit increasing returns to scale, i.e., H(q)/q > Hq(q) (see
also Hwang, Mai and Shieh, 2007) . It simply implies that in equilibrium all firms 
produce on the downward sloping part of the average cost curve under Cournot-Nash 
competition with free entry.    
     This completes our modeling of the basic framework for studying the effects of a
specific tax on the oligopolistic firm’s production and location decisions. 
3. Effects of Specific Taxes on Production and Location Decisions 
We are now in a position to examine the effects of a change in the specific tax rate on the
optimum output and location.  Totally differentiating equations (10)-(13) and applying 
Cramer’s rule, we obtain the following results. 
(∂θ/∂t) = (-1/D4)PQQq3Πθhch{[H(q)/q] – Hq} (17) 
(∂h/∂t) = (1/D4)PQQq3Πθθch{[H(q)/q] – Hq} (18) 
(∂q/∂t) = (-D2/D4)PQQq3  (19) 
(∂N/∂t) = (q/D4)(D2{[2PQ + PQQq – c(.)Hqq] + (N - 1)PQQq} – ΠθθΠqh 2) (20) 
(∂Q/∂t) = N(∂q/∂t) + q(∂N/∂t) 
= (q/D4){D2q[2PQ – c(.)Hqq] – ΠθθΠqh 2} (21) 
where Πθh = - chθH(q), Πθθ = - cθθH(q), Πqq = (N + 1)PQ + NPQQq – cHqq, Πq = PQq(N – 
1), Πqh = ch{[H(q)/q] – Hqq}, D2 = ΠθθΠhh - Πθh 2 and D4 is the relevant Hessian
determinant.  It should be noted that Πθθ < 0, D2 > 0 and D4 > 0 by the stability
conditions, ch < 0 can be seen from equation (12) and [H(q)/q] – Hq > 0 is due to 
increasing returns to scale. 
It is clear that the signs of (∂θ/∂t), (∂h/∂t) and (∂q/∂e) crucially depend upon the shape 
of market demand function.  In the case where the market demand function is linear, i.e., 
4
 
  
 
 
 
  
 
 
 
 
 
 
     
    
  
 
 
 
 
                                                                                
                                                                                 
 
 
PQQ = 0. From (17) – (19), we obtain (∂q/∂t) = 0, (∂θ/∂t) = 0 and (∂h/∂t) = 0. Thus, we
can conclude that 
Proposition 1. The optimum output and location of an oligopolistic firm is independent 
of a change in the specific tax if the demand function is linear.
The economic interpretation behind Proposition 1 is given as follows.  A change in the 
specific tax does not change the slope of the demand curve at any output level but will 
increase the output price in equilibrium for the oligopolistic firms to break even.  In the 
case where the demand function is linear, i.e., PQQ = 0, a higher output price will not alter 
the slope of demand curve and so the required tangency between demand curve and 
average cost curve occurs at the same output level for each firm, i.e., (∂q/∂t) = 0. Since 
the output per firm remains unchanged, the optimum location will remain the same. 
     Next, we consider the case where the demand function is not linear, i.e., PQQ ≠ 0. 
Since the signs of PQQ and Πθh can not a priori be determined, the signs of (∂q/∂t), (∂θ/∂t) 
and (∂h/∂t) in (17) - (19) are ambiguous.  However, from (18) and (19), we can obtain  
(∂q/∂t) < ( >) 0, as PQQ > (<) 0 (22) 
(∂h/∂t) > (<) 0, as PQQ > (<) 0 (23) 
Thus, we can conclude that 
Proposition 2.  An increase in the specific tax will increase (decrease) the output of an 
oligopolistic firm and will move its plant location closer to (farther away from) the CBD 
if the demand function is concave (convex).
The impact of the specific commodity tax on the output per firm is consistent with B1. 
The economic intuition underlying Proposition 2 is given as follow.  An increase in the 
specific tax rate does not change the slope of the demand curve at any output level but 
will increase the output price in equilibrium for firms to break even.  In the case where 
the demand function is concave (i.e., PQQ < 0), a higher output price decreases the 
absolute value of the slope of the demand curve and so the point of tangency between 
demand curve and average curve occurs at a larger output level for each firm.  Since the 
production function exhibits increasing returns to scale, the quantity of inputs, m1 and m2, 
per unit of output declines, then the resources pull decreases while the market pull 
increases. As a result, the optimum location moves towards the CBD.  In the case where 
the demand function is convex (i.e., PQQ > 0), the opposite applies. 
Next, we turn to the effect of a change in the specific tax on equilibrium number of 
undifferentiated oligopolistic firms.  From equation (20), we can see the sign of (∂N/∂t)
can not be a priori determined because [2PQ + PQQq – c(.)Hqq]D2 < 0, - ΠθθΠqh 2 > 0 and 
the sign of (N-1)PQQq2 can be either positive or negative depending on the shape of 
demand function.  We can show 
(∂N/∂t) > (<) 0, as - D2{[2PQ + PQQq – c(.)Hqq] + (N - 1)PQQq} > (<) – ΠθθΠqh 2  (24) 
Thus, we can conclude that 
5
 
  
  
 
     
 
 
 
  
 
 
 
 
 
      
 
 
 
 
 
 
 
 
 
Proposition 3. An increase in the specific commodity tax may increase the number of 
undifferentiated firms even if the demand function is linear or concave. 
This result is significantly different from that of Besley (1989, p. 363) in the non-spatial 
setting. The different results are due to the location effect, ΠθθΠqh 2 = - cθθH(q) 
{ch[H(q)/q]-Hq}2 . It shows that the firm will change its plant location after the change of 
the specific commodity tax.  If the location effect dominates the output effect, the number 
of firms may increase. 
Finally, we consider the effect of an increase in the commodity specific tax on the 
total output of oligopoly. From equation (21), we obtain 
(∂Q/∂t) < (>) 0, as - D2q[2PQ – c(.)Hqq] > (< ) – ΠθθΠqh 2  (25) 
Thus, we can conclude that 
Proposition 8. An increase in the specific tax may increase total output of 
undifferentiated oligopoly.
This result is also significantly different from that of Besley (1989, p. 363).  Once again 
the different results are due to the location effect.   
4. Concluding Remarks 
We examine the impact of a specific commodity tax on the production and plant location 
decisions of undifferentiated oligopolistic firms with free entry.  In the case where the 
demand function is linear, we show that an increase in the specific commodity tax does 
not change the location decision and output of an oligopolistic firm. In the case where 
the demand function is not linear, we show that an increase in the specific tax will cause 
each firm’s output to rise (fall) and move the plant location closer to (farther away from)
the output market if the demand function is concave (convex).  These results indicate 
that B1 holds in the Weber triangular location model. 
In the case where the demand function is linear or concave, we show that an increase
in the specific commodity tax may increase the number of firms and total output of 
oligopoly. This result is significantly different from B2 and B3.  It indicates that the 
location decision has very important influence on the impact of a change in the specific 
commodity tax on the number of firms and total output of oligopoly with free entry. 
References
Besley, T. (1989) “Commodity taxation and imperfect competition: a note on the effects 
of entry” Journal of Public Economics 40, 359-67. 
Hanoch, G. (1975) “The elasticity of scale and the shape of average costs” American 
Economic Review 65, 492-497. 
6
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Hwang, H., Mai, C. C. and Shieh, Y. N. (2007) “Production-location decision and free
entry oligopoly: a correction” Journal of Urban Economics 68, 576-79. 
Kusumoto, S. I. (1984) “On a foundation of the economic theory of location - transport 
distance vs. technological substitution” Journal of Regional Science 24, 249-270. 
Mai, C. C. and Hwang, H. (1992) “Production-location decision and free entry oligopoly” 
Journal of Urban Economics 31, 252-71. 
Seade, J. (1980) “On the effects of entry” Econometrica 48, 479-89. 
Takayama, A. (1993) Analytical Methods in Economics, University of Michigan Press: 
Ann Arbor. 
Weber, A. (1929) Theory of Location of Industry, translated by C.J. Friedrich, University
of Chicago Press: Chicago. 
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The Effects of Specific Commodity Taxes on Output and Location of Free Entry Oligopoly

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